hard problem
by Cobedangiu, Apr 21, 2025, 1:51 PM
Let 
and 
. Prove that:
and 
. Prove that:
Bounding number of solutions for floor function equation
by Ciobi_, Apr 2, 2025, 12:39 PM
Let 
be a positive integer. Consider the following equation: ![\[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]](//latex.artofproblemsolving.com/8/2/9/8299b3ceb1650418c0a373536cb1d2931900efd9.png)
a) For 
, solve the given equation in 
.
b) Prove that, for any, the equation has at most 
real solutions.
be a positive integer. Consider the following equation: ![\[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]](http://latex.artofproblemsolving.com/8/2/9/8299b3ceb1650418c0a373536cb1d2931900efd9.png)
a) For 
, solve the given equation in 
.b) Prove that, for any
, the equation has at most 
real solutions.
satisfy that 
.
Function on positive integers with two inputs
by Assassino9931, Jan 27, 2025, 10:03 AM
The function 
is such that 
for any positive integers 
. Assume there exists a positive integer 
such that 
for all positive integers 
. Determine all possible values of 
.
is such that 
for any positive integers 
. Assume there exists a positive integer 
such that 
for all positive integers 
. Determine all possible values of 
.This post has been edited 1 time. Last edited by Assassino9931, Jan 27, 2025, 10:06 AM
Problem 1
by SpectralS, Jul 10, 2012, 5:24 PM
Given triangle 
the point 
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
at 
, and to the lines 
and 
at 
and 
, respectively. The lines 
and 
meet at 
, and the lines 
and 
meet at 
Let 
be the point of intersection of the lines 
and , and let 
be the point of intersection of the lines 
and 
Prove that is the midpoint of 
(The excircle of opposite the vertex 
is the circle that is tangent to the line segment , to the ray beyond 
, and to the ray beyond 
.)
Proposed by Evangelos Psychas, Greece
the point 
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
at 
, and to the lines 
and 
at 
and 
, respectively. The lines 
and 
meet at 
, and the lines 
and 
meet at 
Let 
be the point of intersection of the lines 
and , and let 
be the point of intersection of the lines 
and 
Prove that is the midpoint of 
(The excircle of
opposite the vertex 
is the circle that is tangent to the line segment , to the ray beyond 
, and to the ray beyond 
.)Proposed by Evangelos Psychas, Greece
APMO 2012 #3
by syk0526, Apr 2, 2012, 3:09 PM
Determine all the pairs 
of a prime number 
and a positive integer 
for which 
is an integer.
of a prime number 
and a positive integer 
for which 
is an integer.
Two circles, a tangent line and a parallel
by Valentin Vornicu, Oct 24, 2005, 10:15 AM
Two circles 
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.Number theory or function ?
by matematikator, Mar 18, 2005, 2:10 PM
Does there exist a function 
such that if 
and 
are distinct rational numbers satisfying 
or 
, then 
? Justify your answer.
Proposed by Dan Brown, Canada
such that if 
and 
are distinct rational numbers satisfying 
or 
, then 
? Justify your answer.Proposed by Dan Brown, Canada
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